GRAVES, JOHN THOMAS (1806–1870), jurist and mathematician, born in Dublin 4 Dec. 1806, was son of John Crosbie Graves, barrister, grandnephew of Richard Graves, D.D. [q. v.], and cousin of Robert James Graves, M.D. [q. v.] After an undergraduate career in Trinity College, Dublin, where he distinguished himself in both science and classics, was a class-fellow and friend of Sir William Rowan Hamilton [q. v.], and graduated B.A. in 1827, he removed to Oxford, where he became an incorporated member of Oriel College, 11 Nov. 1830. Graves proceeded M.A. at Oxford in 1831, and at Dublin in 1832. He was called to the English bar in 1831 as a member of the Inner Temple, having previously (1830) entered the King's Inns, Dublin. For a short time he went the western circuit, and in 1839 he was appointed professor of jurisprudence in London University College in succession to John Austin [q. v.], who finally retired in 1835. Not long after Graves was elected an examiner in laws in the university of London.

The records of Graves's work as a jurist are twelve lectures on the law of nations, reported in the ‘Law Times,’ commencing 25 April 1845, and two elaborate articles contributed to the ‘Encyclopædia Metropolitana’ on Roman law and canon law. He was also a contributor to Smith's ‘Dictionary of Greek and Roman Biography,’ in which, among other articles from his pen, are very full lives of the jurists Cato, Crassus, Drusus, Gaius, and one on the legislation of Justinian. Graves held a high place among the mathematicians of his day in England. In his twentieth year (1826) he engaged in researches respecting exponential functions, which conducted him to important results. They were printed in the ‘Philosophical Transactions’ for 1829 under the title ‘An Attempt to Rectify the Inaccuracy of some Logarithmic Formulæ.’ Of these results one of the principal is the discovery of the existence of two arbitrary and independent integers in the complete expression of an imaginary logarithm. He considered that thus a solution was afforded for various difficulties that had formerly perplexed mathematicians, and that he had elucidated the subject of the logarithms of negative and imaginary quantities, which at different periods had occasioned controversies between Leibnitz and John Bernoulli, Euler, and D'Alembert. His claim to independent discovery and priority of printed publication was undisputed, though M. Vincent of Lille claimed to have arrived in 1825 at similar results, which, however, were not published by him till 1832. The conclusions announced by Graves were not at first accepted by Peacock, who referred to them in his well-known ‘Report on Algebra,’ nor by Sir John Herschel. Graves accordingly communicated to the British Association in 1834 (see the Report for that year) a defence and explanation of his discovery, and in the same report is contained a paper by Sir W. Rowan Hamilton, in which he comes to the support of his friend, giving the conclusions Graves had arrived at the fullest confirmation. This paper bears as its title ‘On Conjugate Functions or Algebraic Couples, as tending to illustrate generally the Doctrine of Imaginary Quantities, and as confirming the Results of Mr. Graves respecting the existence of Two independent Integers in the complete expression of an Imaginary Logarithm.’ It was an anticipation, as far as publication was concerned, of an extended memoir, which had been read by Hamilton before the Royal Irish Academy on 24 Nov. 1833, ‘On Conjugate Functions or Algebraic Couples,’ and subsequently published in the seventeenth volume of the ‘Transactions’ of the Royal Irish Academy. To this memoir were prefixed ‘A Preliminary and Elementary Essay on Algebra as the Science of Pure Time,’ and some ‘General Introductory Remarks.’ In the concluding paragraphs of each of these three papers Hamilton carefully acknowledges that it was ‘in reflecting on the important symbolical results of Mr. Graves respecting imaginary logarithms, and in attempting to explain to himself the theoretical meaning of those remarkable symbolisms,’ that he was conducted to ‘the theory of conjugate functions, which, leading on to a theory of triplets and sets of moments, steps, and numbers,’ became the foundation of his future remarkable contributions to algebraical science, culminating in the discovery of quaternions. For many years Graves and Hamilton maintained an active correspondence, in which they vied with each other in endeavours to carry into space a full and coherent interpretation of imaginaries. Graves worked as having for his aim the perfecting of algebraic language; Hamilton had persistently in view the higher object of arriving at the meaning of the science and its operations. These con- joint labours bore their great fruit in 1843, when Hamilton discovered quaternions, and it was to Graves that he made on 17 Oct. his first written communication of the discovery. In his preface to the ‘Lectures on Quaternions’ and in a ‘prefatory letter’ to a communication to the ‘Philosophical Magazine’ for December 1844 will be found ample acknowledgments of his indebtedness to his friend for stimulus and suggestion. Graves modestly disclaimed the credit of suggestion, and continued to be a sympathetic companion of the great mathematician in all his future work. Soon after the communication to him of the discovery of quaternions Graves employed himself in extending to eight squares Euler's theorem that the sum of four squares multiplied by the sum of four squares gives a product which is also the sum of four squares, and went on to conceive a theory of octaves analogous to Hamilton's theory of quarternions, introducing four imaginaries, additional to Hamilton's i j k, and conforming to ‘the law of the modulus.’ This he imparted to Hamilton, in whom it excited great interest, but on account of its imperfection in the combination of factors it had to resign competition with quaternions as a working calculus. The same is to be said of a pure-triplet system founded on the roots of positive unity, which about this time Graves devised in remarkable coincidence with his brother, Professor Charles Graves, now bishop of Limerick. He afterwards stimulated Sir W. Rowan Hamilton in the study of polyhedra, and received in consequence from him the first intimation of the discovery of the icosian calculus, to which Hamilton was conducted by that study. In addition to the publications already mentioned Graves contributed to the ‘Philosophical Magazine’ for April 1836 a paper ‘On the lately proposed Logarithms of Unity in reply to Professor De Morgan,’ and in the ‘London and Edinburgh Philosophical Magazine’ for the same year a ‘postscript’ entitled ‘Explanation of a Remarkable Paradox in the Calculus of Functions, noticed by Mr. Babbage.’ To the same periodical he contributed in September 1838 ‘A New and General Solution of Cubic Equations;’ in 1839 a paper ‘On the Functional Symmetry exhibited in the Notation of certain Geometrical Porisms, when they are stated merely with reference to the arrangement of points;’ and in April 1845 a paper on the ‘Connection between the General Theory of Normal Couples and the Theory of Complete Quadratic Functions of Two Variables.’ A subsequent number contains a contribution ‘On the Rev. J. G. MacVicar's Experiment on Vision,’ and the ‘Report’ of the Cheltenham meeting in 1856 of the British Association contains abstracts of papers communicated by him ‘On the Polyhedron of Forces’ and ‘On the Congruence nx≡n + 1 (mod. p.).’

Graves was one of the committee of the Society for the Diffusion of Useful Knowledge. In 1839 he was elected a member of the Royal Society, and he subsequently sat upon its council. He was also a member of the Philological Society and of the Royal Society of Literature. For many years he occupied himself in forming a collection of mathematical works of all ages and countries. This portion of his library he bequeathed to University College, London, in remembrance of his former connection as professor with that institution. From the preface to the catalogue of the library of University College the following extract is taken as showing the extent and value of this bequest: ‘The Graves Library is a most valuable collection of more than ten thousand books and about half as many pamphlets. … Perhaps no private scholar has ever formed a mathematical library so nearly complete. Many of the books are very rare, some probably unique, and about one half of the whole collection is in handsome bindings.’ In 1846 Graves was appointed an assistant poor-law commissioner, and in the next year, under the new Poor Law Act, one of the poor-law inspectors of England and Wales. He married in 1846 a daughter of William Tooke, F.R.S., and died without issue on 29 March 1870 at Cheltenham, soon after his resignation of his office.

[An obituary notice of Graves is prefixed to the Proceedings of the Royal Society, vol. xix., and the University College (London) Gazette, vol. i. No. 12, contains a memoir, which concludes with a sketch of his personal character. For additional particulars reference may be made to the Life of Sir William Rowan Hamilton, 3 vols. Dublin, 1882, 1885, 1888.]